Representation of permutations as products of cycles of fixed length

1976 ◽  
Vol 22 (3) ◽  
pp. 321-331 ◽  
Author(s):  
Marcel Herzog ◽  
K. B. Reid
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Fixed Length ◽  
Minimum Number

AbstractWe study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.

scholarly journals A Turán Type Problem Concerning the Powers of the Degrees of a Graph

10.37236/1525 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Degree Sequence ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Type Problem ◽  
Maximum Value ◽  
Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

Almost Safe Group Testing with Few Tests

1994 ◽  
Vol 3 (3) ◽  
pp. 411-419
Author(s):  
Andrzej Pelc
Keyword(s):  
Lower Bounds ◽  
Group Testing ◽  
Upper And Lower Bounds ◽  
Probability Functions ◽  
Minimum Number ◽  
Location Strategies

In group testing, sets of data undergo tests that reveal if a set contains faulty data. Assuming that data items are faulty with given probability and independently of one another, we investigate small families of tests that enable us to locate correctly all faulty data with probability converging to one as the amount of data grows. Upper and lower bounds on the minimum number of such tests are established for different probability functions, and respective location strategies are constructed.

Composite Thermoelectrics - Exact Results and Calculational Methods

1991 ◽  
Vol 234 ◽  
Author(s):  
David J. Bergman ◽  
Ohad Levy
Keyword(s):  
Quality Factor ◽  
Computer Simulations ◽  
Lower Bounds ◽  
Theoretical Study ◽  
Transport Coefficients ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Thermoelectric Transport

ABSTRACTA theoretical study of composite thermoelectric media has resulted in the development of a number of simple approximations, as well as some exact results. The latter include exact upper and lower bounds on the bulk effective thermoelectric transport coefficients of the composite and upper bounds on the bulk effective thermoelectric quality factor Ze. In particular, as a result of some exact theorems and computer simulations we conclude that Ze can never be greater than the largest value of Z in the different components that make up the composite.

scholarly journals On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

2020 ◽  
Vol 36 (36) ◽  
pp. 124-133
Author(s):  
Shinpei Imori ◽  
Dietrich Von Rosen
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Degrees Of Freedom ◽  
Generalized Inverse ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Identity Matrix ◽  
Wishart Matrix ◽  
The Mean ◽  
Scale Matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution.

The Thickness of the Cartesian Product of Two Graphs

2016 ◽  
Vol 59 (4) ◽  
pp. 705-720
Author(s):  
Yichao Chen ◽  
Xuluo Yin
Keyword(s):  
Lower Bounds ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Minimal Graph ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

AbstractThe thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

scholarly journals TRIPLE CROSSING NUMBER OF KNOTS AND LINKS

2013 ◽  
Vol 22 (02) ◽  
pp. 1350006 ◽  
Author(s):  
COLIN ADAMS
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Upper And Lower Bounds ◽  
Minimum Number ◽  
Bracket Polynomial ◽  
Knots And Links

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c3(K) for a variety of knots and links. We then use c3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to cn(K).

On the estimates of the values of various widths of classes of functions of two variables in the weight space $L_{2,\gamma}(\mathbb{R}^2)$, $\gamma=\exp(-x^2-y^2)$

2020 ◽  
Vol 17 (1) ◽  
pp. 95-115
Author(s):  
Sergey Vakarchuk ◽  
Mihail Vakarchuk
Keyword(s):  
Differential Operator ◽  
Lower Bounds ◽  
Modulus Of Continuity ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Weight Space ◽  
Generalized Modulus Of Continuity ◽  
Exact Results ◽  
Functions Of Two Variables

For the classes of functions of two variables $W_2(\Omega_{m,\gamma},\Psi)=\{ f \in L_{2,\gamma}(\mathbb{R}^2) : \Omega_{m,\gamma}(f,t) \leqslant \Psi(t) \, \forall t \in (0,1)\}$, $m \in \mathbb{N}$, where $\Omega_{m,\gamma}$ is a generalized modulus of continuity of the $m$-th order, and $\Psi$ is a majorant, the upper and lower bounds for the ortho-, Kolmogorov, Bernstein, projective, Gel'fand, and linear widths in the metric of the space $L_{2,\gamma}(\mathbb{R}^2)$ are found. The condition for a majorant under which it is possible to calculate the exact values of the listed extreme characteristics of the optimization content is indicated. We consider the similar problem for the classes $W^{r,0}_2(\Omega_{m,\gamma},\Psi)=L^{r,0}_{2,\gamma}(D,\mathbb{R}^2) \cap W^r_2(\Omega_{m,\gamma},\Psi)$, $r,m \in \mathbb{N}$, $\big(D=\frac{\displaystyle \partial^2}{\displaystyle \partial x^2} + \frac{\displaystyle \partial^2}{\displaystyle \partial y^2} -2x\frac{\displaystyle \partial}{\displaystyle \partial x} -2y\frac{\displaystyle \partial}{\displaystyle \partial y}$ being the differential operator$\big)$. Those classes consist of functions $f \in L^{r,0}_{2,\gamma}(\mathbb{R}^2)$ whose Fourier--Hermite coefficients are $c_{i0}(f) = c_{0j}(f)=c_{00}(f)=0$ $\forall i, j \in \mathbb{N}$. The $r$-th iterations $D^rf = D(D^{r-1}f)$ $(D^0f \equiv f)$ belong to the space $L_{2,\gamma}(\mathbb{R}^2)$ and satisfy the inequality $\Omega_{m,\gamma}(D^rf,t) \leqslant \Psi(t)$ $\forall t \in (0,1)$. On the indicated classes, we have determined the upper bounds (including the exact ones) for the Fourier--Hermite coefficients. The exact results obtained are specified, and a number of comments regarding them are given.

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